Framework and diagnostics
ANU BDSI
workshop Generalised Linear Models
Emi Tanaka
Biological Data Science Institute
27th September 2024
Generalised linear models
Genralised linear models (GLMs) has three components:
- Assume that Y_i is a member of an exponential family (e.g. Normal/Gaussian, Binomial, Poission, Gamma, etc).
- A linear predictor \eta_i = \beta_0 + \beta_1 x_{i1} + \ldots + \beta_k x_{ik}.
- A (smooth and invertible) link function g(\cdot) which transforms the expectation of the response variable \mu_i = E(Y_i) to the linear predictor: g(\mu_i) = \eta_i = \beta_0 + \beta_1 x_{i1} + \ldots + \beta_k x_{ik}.
Link functions
| Identity |
\mu_i |
\eta_i |
| Log |
\log(\mu_i) |
\exp(\eta_i) |
| Inverse |
1/\mu_i |
1/\eta_i |
| Inverse-squared |
1/\mu_i^2 |
1/\sqrt{\eta_i} |
| Square-root |
\sqrt{\mu_i} |
\eta_i^2 |
| Logit |
\log(\mu_i/(1-\mu_i)) |
\exp(\eta_i)/(1+\exp(\eta_i)) |
| Probit |
\Phi^{-1}(\mu_i) |
\Phi(\eta_i) |
| Complementary log-log |
\log(-\log(1-\mu_i)) |
1-\exp(-\exp(\eta_i)) |
where \Phi(\cdot) is the cumulative distribution function of the standard normal distribution.
Canonical link
| Normal |
Identity |
(-\infty, \infty) |
\psi |
| Binomial |
Logit |
[0, 1] |
\mu_i(1-\mu_i) |
| Poisson |
Log |
[0, \infty) |
\mu_i |
| Gamma |
Inverse |
(0, \infty) |
\psi\mu_i^2 |
| Inverse-Gaussian |
Inverse-squared |
(0, \infty) |
\psi\mu_i^3 |
where \psi is the dispersion parameter, \eta_i is the linear predictor, and \mu_i is the expectation of Y_i. In the binomial family, n_i is the number of trials.
